A Compelling Formula Indicating the Existence of Ultra-low Energy Levels in the Hydrogen Atom

Einstein's energy-momentum relationship, which holds in an isolated system in free space, cannot be applied to an electron in a hydrogen atom where potential energy is present. The author has previously derived an energy-momentum relationship which holds inside a hydrogen atom. Using an argument of Dirac, a solution is obtained for the positive and negative energy levels based on this relationship too. Even if the energy is described on an absolute scale, the electrons at negative energy levels have negative mass. The author has adopted the name "dark hydrogen atom (DHA)" for this type of unknown material composed of an electron with negative mass and a proton (atomic nucleus) with positive mass. The mass of a DHA is almost the same the mass of an ordinary hydrogen atom. However, a DHA is extremely small compared to an ordinary hydrogen atom, and thus can achieve states of ultra-high density. The author has pointed out that these DHAs, and the hydrogen molecules and other materials formed from them, are the true nature of the unknown material known as "dark matter" whose true nature is currently regarded as unknown. Recently, the author has derived a previously unknown formula for the energy levels of the hydrogen atom. According to this formula, physical quantities of DHA are involved in the energy levels of a hydrogen atom. This paper derives this formula for the energy levels of a hydrogen atom using another method. The author believes this formula to be strong evidence of the existence of DHAs.